outside Scaling Statistical Physics
Robert Gilmore
Mechanical Similarity: The classical Newtonian equation of motion for a particle of mass m in the presence of a potential V (x) is m d2 x = -V (x) dt2
Assume that under a scaling transformation, the mass scales with a factor (i.e., m m), x x, t t. Assume also that the potential is homogeneous of degree k: V (x) k V (x) Under this scaling transformation show that the equation of motion transforms to (1 1 -2 ) m d2 x = - k-1 V (x) dt2
a. Show that the scaled equation is identical to the original provided 1 2-k -2 = 1. b. Set = 1. Show that trajectories are invariant under the scaling transformation with 2 = 2-k . Show that in the cases k = -1, k = 0, k, = +1, k = +2 the following scaling results hold: k -1 0 +1 +2 Potential Type Coulomb No Force Local Gravitational Potential Harmonic Oscillator transformation 2 = 3 2 = 2 2 = 1 2 = 0
The first line is a statement of Kepler's Third Law: for closed planetary orbits, the square of the period ( 2 ) is proportional to the cube of the semiaxis ( 3 ). If R and T are the semiaxis and period of planet P and R and T are the semiaxis and period of planet P , and the two planets P and P have geometrically similar orbits, 3 (R /R)3 = (T /T )2 2 . The second line is a statement of the integral of Newton's second law in the absence of forces in an inertial frame: the distance traveled () is proportional to the time elapsed (). The third line is a statement that in a local gravitational potential of the form V = mgz, the distance fallen increases like the square of the time elapsed. The fourth line is a statement of Hooke's Law: in harmonic motion the period () is independent of the size of the orbit. c. Fix = 1 and construct a table relating the mass and orbital scale under the four forces described in the table above. d. Fix = 1 and show that the period scales like M for all homogeneous potentials. Reconcile this result with the well-known result that the period of a planet is independent of its mass in lowest order. 1
e. If the motion is bounded for all times, show 2 T = xV (x) = kV (x) where T is the kinetic energy. This is the Virial Theorem for homogenoeous potentials. f. Show that the kinetic energy scales like 2 -2 = k (use a.). Since the potential energy scales the same way, the total energy has this scaling property. g. that Assume the potential can also be scaled: V (x) V (x). Use this scaling to determine how long it takes a planet to orbit a star whose mass is twice that of our sun if the planet is the same distance from its star as we are from the sun ( 2 = 1). Sophisticated Similarity: Assume that the dynamics of a system are derivable from an action principle: for example, the Euler-Lagrange equations are derived from the variation of an action: L(x, x)dx = 0. Show that if a scaling transformation leaves the Lagrangian invariant up to an overall scaling factor, the trajectories will scale under this transformation. Quantum Similarity: Scaling in the quantum theory works in basically the same way. We introduce the scaling factors , , , as defined above for p2 m, x, t, V . We also note that the hamiltonian H = 2m + K|q|k depends on p. In order to preserve the canonical commutation relations [pr , qs ] = ( /i)r,s , h momenta must scale contravariantly to the coordinates: p -1 p. These transformations produce a new hamiltonian H = 1 p2 + k K|q|k 2 2m or 2 H = p2 + k+2 K|q|k 2m
If the condition k+2 = 1 is satisfied, the original hamiltonian H and the new hamiltonian H are proportional, with H = ( 2 )-1 H where = ()-1/k+2 . The result is that lengths scale like = ()-1/k+2 and energies scale like -k/k+2 2/k+2 . Time scales contravariantly to the energy, like 1/Energy. In a coulomb potential, k = -1. I...